Samenvatting
For large systems of ordinary differential equations (ODEs), some components
may show a more active behavior than others. To solve such problems nu
merically, multirate integration methods can be very efficient. These methods
enable the use of large time steps for slowly varying components and small steps
for rapidly varying ones. In this thesis we design, analyze and test multirate
methods for the numerical solution of ODEs.
A selfadjusting multirate time stepping strategy is presented in Chapter 1.
In this strategy the step size for a particular system component is determined
by the local temporal variation of this solution component, in contrast to the
use of a single step size for the whole set of components as in the traditional
methods. The partitioning into different levels of slow to fast components is
performed automatically during the time integration. The number of activity
levels, as well as the component partitioning, can change in time. Numerical
experiments confirm that with our strategy the efficiency of time integration
methods can be significantly improved by using large time steps for inactive
components, without sacrificing accuracy.
A multirate scheme, consisting of the ¿method with one level of temporal
local refinement, is analysed in Chapter 2. Missing component values, required
during the refinement step, are computed using linear or quadratic interpolation.
This interpolation turns out to be important for the stability of the multirate
scheme. Moreover, the analysis shows that the use of linear interpolation can
lead to an order reduction for stiff systems. The theoretical results are confirmed
in numerical experiments.
Two multirate strategies, recursive refinement and the compound step strat
egy are compared in Chapter 3. The recursive refinement strategy has somewhat
larger asymptotic stability regions than the compound step strategy. The com
pound step strategy, by avoiding the extra work of doing the macro step for all
the components, looses some stability properties compared to the recursive re
finement strategy. It can also lead to more complex algebraic implicit systems,
which are difficult to solve numerically.
The construction of higherorder multirate Rosenbrock methods is discussed
in Chapter 4. Improper treatment of stiff source terms and use of lowerorder
interpolants can lead to an order reduction, where we obtain a lower order of
consistency than for nonstiff problems. We recommend a strategy of avoidance
of the order reduction for problems with a stiff source term. A multirate method
based on the fourthorder Rosenbrock method RODAS and its thirdorder dense
output has been designed. This multirate RODAS method has shown very good
results in numerical experiments, and it is clearly more efficient than other
considered multirate methods in these tests.
Explicit multirate and partitioned RungeKutta schemes for semidiscrete
hyperbolic conservation laws are analysed in Chapter 5. It appears that, for the
considered class of multirate methods, it is not possible to construct a multirate
scheme which is both locally consistent and massconservative. The analysis
shows that, in spite of local inconsistencies, global convergence is still possible
in all grid points.
Originele taal2  Engels 

Kwalificatie  Doctor in de Filosofie 
Toekennende instantie 

Begeleider(s)/adviseur 

Datum van toekenning  15 jan 2008 
Plaats van publicatie  Amsterdam 
Uitgever  
Gedrukte ISBN's  9061965438 
Status  Gepubliceerd  2008 
Vingerafdruk Duik in de onderzoeksthema's van 'Multirate numerical integration for ordinary differential equations'. Samen vormen ze een unieke vingerafdruk.
Citeer dit
Savcenco, V. (2008). Multirate numerical integration for ordinary differential equations. Universiteit van Amsterdam.